New York Journal of Mathematics
New York J. Math. 3A (1997) 15–30.
Convergence of the p-Series
for Stationary Sequences
I. Assani
Abstract. Let (Xn ) be a stationary sequence. We prove the following
(i) If the variables (Xn ) are iid and E(|X1 |) < ∞ then
lim
(p − 1)
p→1+
(ii)
X
∞
n=1
|Xn (x)|p
np
1/p
= E(|X1 |), a.e.
If Xn (x) = f (T n x) where (X, F , µ, T ) is an ergodic dynamical system, then
lim
p→1+
(p − 1)
X
∞ n=1
f (T n x)
n
p 1/p
Z
=
f dµ
a.e. for f ≥ 0, f ∈ L log L.
Furthermore the maximal function,
sup (p−1)1/p
1<p<∞
X
∞ n=1
f (T n x)
n
p 1/p
is integrable for functions,f ≥ 0, f ∈ L log L.
These limits are linked to the maximal function N ∗ (x) = k(
Xn (x)
)k1,∞ .
n
Contents
1. Introduction
2. Convergence of the p-series for iid sequences
3. Convergence of the p-series for ergodic stationary sequences
References
15
18
23
30
1. Introduction
Let Zn be a sequence of independent, identically distributed random variables
and (an ) a sequence of positive real numbers. The a.e. convergence of the weighted
averages
(∗)
PN
n=1
an Zn
An
,
Received June 9, 1997.
Mathematics Subject Classification. 28D05, 60F15, 60G50.
Key words and phrases. pSeries, maximal function, iid random variables and stationary sequences.
c
1997
State University of New York
ISSN 1076-9803/97
15
16
I. Assani
PN
where An = n=1 an , has been characterized by B. Jamison, S. Orey and W. Pruitt
([JOP]). They proved that the condition
(0)
sup
n
Ñn
<∞
n
ak
where Ñn = #{k : A
≥ n1 } is necessary and sufficient for the a.e. convergence of
k
the weighted averages (∗) to E(Z1 ). In [A1], interested by the a.e. convergence (y)
of averages of the form
PN
n
n=1 Xn (x)g(S y)
,
N
we considered the maximal function N ∗ (x) = supn Nnn(x) where Nn (x) = #{k :
Xk (x)
≥ n1 }, (Xk ≥ 0). We proved the following:
k
(1) If Xn are iid random variables and E(|X1 |) < ∞ then N ∗ (x) is finite a.e.
(2) If the Xn are given by an ergodic dynamical system (i.e., Xn (x) = f (T n x)
where (X, F, µ, T ) is an ergodic dynamical system and f a measurable nonnegative function) then for all p, 1 < p < ∞ there exists a finite constant
Cp such that
Z
Cp
∗
(∗∗)
µ{x : N (x) > λ} ≤ p
|f |p dµ
for all λ > 0.
λ
Furthermore for all p, 1 < p < ∞, for all f ∈ Lp+ we have lim Nnn(x) =
n→∞
R
f dµ a.e.
(A closer inspection of the proof of (∗∗) shows that the constant Cp is of the
C
form p−1
where C is an absolute constant independent of p.)
If 0 < p < ∞, and (xi )i≥1 is a sequence of nonnegative real numbers, set
1/p
sup λp #{i ≥ 1; |xi | > λ}
k(xi )kp,∞ =
λ>0
.
It is easily seen that for r < p
k(xi )kp,∞ ≤
X
|xi |p
1/p
≤
i
p
p−r
1/p
k(xi )kr,∞
(cf. [SW]). In particular, for all p, 1 < p ≤ 2 we have
(3)
(p − 1)1/p
X
|xi |p
1/p
≤ p1/p k(xi )k1,∞ .
i
As k(xi )k1,∞ ∼ supn #{k:xkn≥1/n} , for bounded sequences the previous inequality
applied pointwise to a stationary sequence (Xn ) of integrable functions gives us not
only the existence of the p-series
(p − 1)
1/p
X
i
Xi (x) p
|
|
i
1/p
for all p, 1 < p ≤ ∞,
Convergence of the p-Series for Stationary Sequences
17
but also the inequality
sup (p − 1)1/p
(4)
X
1<p<∞
i
|
Xi (x) p
|
i
1/p
≤ 2k(
Xi (x)
)k1,∞
i
if k( Xii(x) )k1,∞ < ∞. The inequality (4) and some of our previous results suggest
the study of the limit when p tends to 1+ of the series
(p − 1)
1/p
X
∞
Xi (x) p
|
|
i
i=1
1/p
.
Definition. Let (Xn ) be a stationary sequence of integrable functions. The p series
associated to this sequence is the a.e. series (when it exists):
(p − 1)
1/p
X
∞
Xi (x) p
|
|
i
i=1
1/p
.
In this note, using an elementary lemma on sequence of real numbers, we will
show that for (Xn ) iid with E(|X1 |) < ∞ the p-series
(p − 1)1/p
X
∞
i=1
|
Xi (x) p
|
i
1/p
converges a.e. to E(|X1 |)
when p tends to 1+ .
The same argument shows that the p series
(p − 1)
1/p
X
∞ QH
j=1 Xj,i (xj ) p 1/p
i
converges a.e. to
i=1
H
Y
E(|Xj,1 |)
j=1
where (Xj,n )n are iid random variables satisfying the condition E(|Xj,1 |) < ∞, and
the variables xj are selected in a universal way specified in [A1].
P
1/p
∞
Xi (x) p
We can remark that for each p the function Gp (x) = (p−1)1/p
i=1 | i |
is not integrable, as Gp (x) ≥ (p − 1)1/p supi | Xii(x) |, and for (Xi ) iid with
E(|X1 | log |X1 |) = ∞, the function supi | Xii(x) | is not integrable, as shown by D.
Burkholder in [B]. So F ∗ (x) = sup Gp (x) is a supremum of nonintegrable func1<p<∞
tions. This makes the handling of the function F ∗ (x) somewhat delicate.
In the second part of this note we will focus on the ergodic stationary case. We
will consider an ergodic dynamical system (X, F, µ, T ) and a nonnegative measurable function f . Using (2) we will show first that
#{k :
Nn (f )(x)
=
n
f (T k x)
k
n
≥ 1/n}
18
I. Assani
R
converges in L1 norm to f dµ. Then using extrapolation methods we will show
that
f (T k x) < ∞ forf ∈ L(Log L).
(5)
k
1,∞ 1
One of our interests in (5) lies in the following observation: If we denote by
∗
f (T n x)
n∗
a decreasing rearrangement of the sequence
f (T n x)
,
n
then we have
∗
f (T k x) f (T n x)
= sup n
.
k
n∗
n
1,∞
(6)
Hence for f ∈ L(Log L), (6) provides us with some information on the decreasing
n
rate of the sequence f (Tn x) .
Using (5), we will prove that for f ∈ L log L, f ≥ 0,
M1∗ (x)
0
(6 )
= sup (p − 1)
1/p
X
∞ 1<p<∞
n=1
f (T n x)
n
p 1/p
,
and
(7)
lim+ (p − 1)
1/p
X
∞ p→1
n=1
f (T n x)
n
p 1/p
Z
=
f dµ a.e., (µ).
The integrability of M1∗ (x) for f in LLogL extends the results on the integrability of
n
the supn f (Tn x) in the ergodic case. We do not know at the present time if (7) holds
for f ∈ L1 . Finally, in the third part of this paper we will study the connection
between the maximal operators
M1∗ (f )(x) = sup (p − 1)1/p
1<p<∞
X
∞ n=1
f (T n x)
n
p 1/p
, M2∗ (f )(x) = sup
N
N
1 X
f (T n x)
N n=1
and
f (T n x) = N ∗ (f )(x).
n
1,∞
If there is no ambiguity we will simply denote these maximal functions by M1∗ (x),
M2∗ (x) and N ∗ (x).
2. Convergence of the p-series for iid sequences
2.1. The one dimensional case. The next elementary lemma will be useful for
the convergence we are looking for.
Convergence of the p-Series for Stationary Sequences
Lemma 1. Let (xn )n be a sequence of nonnegative numbers such that
#{k :
xk
k
≥ 1/n}
19
xk
→ 0 and
k k
7→ x̄, then
P∞ xn p 1/p
= x̄.
(a) limp→1+ (p − 1)1/p
n=1 ( n )
xk∗
xk
xk∗
(b) If ∗ is a decreasing rearrangement of the sequence ( )k then k · ∗
k
k
k
converges to x̄.
n
Proof. We denote by Rn = {k : xkk ≥ 1/n} and Nn = #{k :
To prove (a) it is enough to show that
X
∞
xn
lim (p − 1)
( )p
n
p→1+
n=1
We can write the series (p − 1)(
P∞
xn p
n=1 ( n ) )
xk
k
≥ 1/n} = #Rn .
= x̄.
in the following way;
X
X
∞
xn p
xn
(p − 1)
( ) = (p − 1)
( )p +
n
n
n=1
n∈R1
X
n∈N∗ \R1
xn
( )p
n
= Ap + Bp .
As lim Ap = 0 we just need to consider Bp = (p − 1)
p→1
P
Xn p
n∈N∗ \R1 ( n ) .
But we have
∞
∞
X
X
Nn+1 − Nn
Nn+1 − Nn
(p − 1)
≤ Bp ≤ (p − 1)
.
p
(n
+
1)
np
n=1
n=1
It is then enough to prove that Bp is squeezed into two terms tending to the
P∞
−Nn
consame limit x̄. We will only prove that the term (p − 1) n=1 Nn+1
np
verges to x̄. The same argument shows the same conclusion for the second term
P∞
n+1 −Nn
(p − 1) n=1 N(n+1)
.
p
We have
(p − 1)
As
Nn
n
∞
∞
X
N1 X Nn (np − (n − 1)p ))
Nn+1 − Nn
=
(p
−
1)
−
+
np
1p
np (N − 1)p
n=1
n=2
∞
(n−1)
N1 X Nn (1 − ( n )p ))
= (p − 1) − p +
1
(n − 1)p
n=2
∞
X
1
Nn
N1
·
.
∼ (p − 1) − p + p
1
n (n − 1)p
n=2
converges to x̄ and
P∞
1
n=2 (n−1)p
lim (p − 1)
p→1+
∼
1
p−1
we conclude that
∞
X
Nn+1 − Nn
= x̄.
np
n=1
20
I. Assani
(b) To obtain the convergence of the sequence k xkk∗∗ to x̄ we can observe that
lim
#{` :
t→∞
x`
`
t
If we take the increasing sequence tk =
decreasing rearrangement of the sequence
≥ 1t }
= x̄.
xk∗
k∗
xk∗ where k∗ is
xk
k we can see that
xk∗
x`
xk∗
xk∗
≥ ∗ }=k· ∗
· #{` :
∗
k
`
k
k
the kth term of the
converges to x̄.
This ends the proof of this lemma.
In this part we only consider sequences Xn of iid nonnegative random variables
such that E(X1 ) < ∞. This assumption can be made in view of the nature of our
p series.
Theorem 2. Let (Xn ) be a sequence of iid nonnegative random variables such that
E(X1 ) < ∞. Then we have
(a)
(b)
1
Xk (x)
Nn (x)
lim
= E(X1 ) a.e. with Nn (x) = # k :
≥
,
n→∞
n
k
n
p 1/p
X
∞ Xn (x)
lim (p − 1)1/p
= E(X1 ), a.e.
n
p→1+
n=1
Proof. By the previous lemma, (b) is an immediate consequence of (a), so we are
left with proving (a).
In our proof of Lemma 1 in [A1], we showed that we have
Xn (x) #{k :
< ∞ a.e., because limn→∞
n 1,∞
Xk (x)
k
n
≥
1
n}
= E(X1 ).
We proved this by noting that
Nn (x) = #{k :
∞
X
1
1
Xk (x)
Xk (x)
≥ }=
≥
.
1 x:
k
n
k
n
n=1
Then we considered
Vn (x) =
∞
X
1
Xk (x)
1
Xk (x)
≥
−µ x:
≥
.
1 x:
k
n
k
n
n=1
Kolmogorov’s inequality for sums of independent random variables leads to the
following inequality for each > 0.
∞
X
n=1
Nn2 (x) − E(Nn2 ) ≥ } < ∞.
µ{
n2
Convergence of the p-Series for Stationary Sequences
21
An application of the Borel-Cantelli lemma gave us
Nn2 (x)
E(Nn2 )
= lim
= E(X1 ).
2
n
n
n2
Then a simple interpolation allowed us to claim that
lim
limn
(8)
Nn (x)
= E(X1 ).
n
But also in [A1], Theorem 3 shows that for each p, 1 < p ≤ ∞ we have
(9)
lim
#{k :
Yk (x)
k
≥ 1/n}
n
n→∞
= E(Y1 )
for (Yn ) sequence of iid random variables where E(|Y1 |p ) < ∞ for some 1 < p ≤ ∞.
We take M a positive constant; using (8) and (9) we get
#{k :
Xk (x)∧M
k
≥ 1/n}
n
#{k : Xkk(x) ≥ 1/n}
≤ lim
n
#{k : Xkk(x) ≥ 1/n}
= lim
n
= E(X1 ).
E(X1 ∧ M ) = lim
n
As limM E(X1 ∧ M ) = E(X1 ) we have obtained a proof of (a) from which (b) now
follows easily.
2.2. The multidimensional case. The previous situation can be extended to a
more general situation. In [A1] we proved the following:
Given H a positive integer and a nonnegative iid sequence (X1,n )n
on the probability measure space (Ω1 , F1 , µ1 ) satisfying the condition
e 1 such that if
E(X1,1 ) < ∞, it is possible to find a set of full measure Ω
e
x1 ∈ Ω1 the following holds:
For all probability measure spaces (Ω2 , F2 , µ2 ) and all nonnegative
iid sequences (X2n )n such that E(X2,1 ) < ∞ it is possible to find a set
e 2 such that if x2 ∈ Ω
e 2 the following holds:
of full measure Ω
For all probability measure spaces (ΩH , FH , µH ) and all iid sequences
(XH,n )n of nonnegative random variables satisfying E(XH,1 ) < ∞ we
e H for which if xH ∈ Ω
e H we have
can find a set of full measure Ω
(10)
limn
#{k :
QH
i=1
Xi,k (xi )
k
n
≥
1
n}
=
H
Y
E(Xi,1 ).
i=1
e i are obtained; they
The difficulty resides in the way those sets of full measure Ω
are independent of the incoming variables (Xj,n ) for j > i.
QH
We want to prove that in (10) we actually have convergence to i=1 E(Xi,1 ).
More precisely we have:
22
I. Assani
Theorem 3. Given H a positive integer and a nonnegative sequence of iid variables (X1n )n on the probability measure space (Ω1 , F1 , µ1 ) satisfying the condition
e 1 such that if x1 ∈ Ω
e 1 the
E(X1,1 ) < ∞, it is possible to find a set of full measure Ω
following holds:
For all probability measure spaces (Ω2 , F2 , µ2 ) and all nonnegative iid sequences
e 2 such
(X2,n )n such that E(X2,1 ) < ∞, it is possible to find a set of full measure Ω
e
that if x2 ∈ Ω2 the following holds:
For all probability measure spaces (ΩH , FH , µ, H) and all iid sequences (XH,n )n
of nonnegative random variables satisfying E(XH,1 ) < ∞ we can find a set of full
e H for which if xH ∈ Ω
e H we have
measure Ω
(11)
lim
#{k :
QH
i=1
Xi,k (xi )
k
≥
n
n
1
n}
=
H
Y
E(Xi,1 )
i=1
and
(12)
X
∞
lim+ (p − 1)
p→1
QH
i=1
n=1
Xi,n (xi )
n
!p 1/p
=
H
Y
E(Xi,1 ).
i=1
Proof. As previously we just need to prove (11) to get (12). We use induction to
prove (11). The result is true for H = 1, as shown in the previous theorem.
QH−1
Let us assume that the result is true for H − 1. Hence if ck = i=1 Xik (xi )
e i we have
where xi ∈ Ω
(13)
lim
#{k :
ck
k
≥
n
n→∞
1
n}
=
H−1
Y
E(Xi,1 ).
i=1
ei ,
The idea of the proof is the same as in Lemma 1 in [A1]. We have for xi ∈ Ω
1 ≤ i ≤ H − 1, (XH,n ) a sequence of nonnegative iid random variables and for all
>0
∞
X
µ xH
(14)
n=1
N 2 (xH ) − E(Nn2 ) ≥ <∞
: n
n2
where
Nn2 (xH ) =
#{k :
ck XH,k (xH )
k
n2
≥ 1/n2 }
.
The inequality (14) is obtained by applying Kolmogorov’s inequality to the series
of independent random variables
∞
X
k=1
1
xH
− µ xH : ck XH,k (xH ) ≥ 1/n .
c X
(x )
k
: k H,k H ≥1/n
k
Convergence of the p-Series for Stationary Sequences
23
The Borel-Cantelli lemma applied to (14) gives us
Nn2 (xH ) − E(Nn2 )
=0
n2
lim
n→∞
As limn→∞
E(Nn2 )
n2
=
QH
i=1
a.e. (xH ).
E(Xi,1 ) we have
H
Nn2 (xH ) Y
=
E(Xi,1 )
n→∞
n2
i=1
lim
a.e. (xH ).
The monotonicity of Nn gives us for p2n ≤ n ≤ (pn+1 )2
Np2n (xH )
N(pn+1 )2 (xH )
N(pn+1 )2 (xH ) (pn+1 )2
Nn (xH )
≤
≤
=
·
.
p2n
p2n
p2n
(pn+1 )2
(pn )2
This last chain of inequalities implies that
H
Np2n (xH ) Y
Nn (xH )
= lim
=
E(Xi,1 )
n→∞
n→∞
n
p2n
i=1
lim
as
p2n
→ 1.
n
3. Convergence of the p-series for ergodic stationary sequences
In this part the sequence Xn will be given by an ergodic dynamical system
(X, F, µ, T ) on a probability measure space (X, F, µ). The sequence is defined by
the relation Xn (x) = f (T n x) where f is a nonnegative integrable function.
Proposition 4. Let (X, F, µ, T ) be an ergodic dynamical system and f a nonnegative integrable function. We have
Z
Nn (f )
= 0, where
−
f
dµ
lim n→∞
n
#{k :
Nn (f )(x)
=
n
1
f (T k x)
k
≥ 1/n}
n
R
Proof. We know that limn→∞ Nnn(f ) = f dµ a.e. for f ∈ Lp+ for some p, 1 <
p ≤ ∞ (see Theorem 3 in [A1]). The difficulty at this level comes from the nature
of the function of f , Nn (f ); the map Nn is not linear nor positively homogeneous.
But we have the following properties:
(A) k Nnn(f ) k∞ ≤ kf k∞ ,
(B) If f, g are nonnegative functions with disjoint support then we have
Nn (f +g)
= Nnn(f ) + Nnn(g) for all n ≥ 1.
n
(C) For all f ≥ 0 integrable functions we have k Nnn(f ) k1 ≤ kf k1 .
24
I. Assani
(A) and (B) are easy to check.
To establish (C) we take f ∈ L1 for whichPwe can find for each nonnegative
(αi )i and sets R(Ai )i such that f ≤
αi 1Ai , Ai ∩ Aj = φ if i 6= j and
Rnumbers
P
αi 1Ai dµ ≤ (1 + ) f dµ. We have
P∞
Nn (
Nn (f )
≤
n
Thus
i=1
αi 1Ai )
n
by monotonicity.
P∞
Nn (f ) ≤ Nn ( i=1 αi 1Ai ) n n
1
1
∞
X N (α 1 ) n
i Ai =
n
i=1
1
∞
X
Nn (αi 1Ai ) .
=
n
1
i=1
by (B)
As
1
(tk x)
1
#{k : Ai k
≥ nα
}
Nn (αi 1Ai )
i
=
n
n
P[nαi ]
1Ai (T k x)
we have
= k=1
n
[nαi ]
X µ(Ai )
Nn (αi 1Ai ) (nαi )µ(Ai )
=
≤
= αi µ(Ai ) .
n
n
n
1
k=1
So
Z
∞
Nn (f ) X
≤
αi µ(Ai ) ≤ (1 + ) f dµ.
n i=1
As is arbitrary we have reached a proof of (C).
We are now in a position to prove Proposition 4.
For each positive real number M we can write f = f ∧ M + gM with f ∧ M and
gM nonnegative functions with disjoint support.
We have
Z
Z
Z
Nn (f ∧ M )
Nn (gM )
Nn (f )
− f dµ =
− f ∧ M dµ +
− gM dµ.
n
n
n
Hence
Z
Z
Nn (f )
≤ lim Nn (f ∧ M ) − (f ∧ M )dµ
−
f
dµ
lim n
n
n
n
1
1
Z
Nn (gM ) + gM dµ.
+ lim n n
1
Convergence of the p-Series for Stationary Sequences
25
By the theorem mentioned
R at the beginning of this proof, associating the a.e. convergence of Nn (fn)(x) to f dµ for functions in Lp for some p, we conclude that
Z
Nn (f ∧ M )
= 0.
−
f
∧
M
dµ
lim n
n
1
Hence
R
As
Z
Z
Nn (f )
≤ 2 gM dµ,
−
f
dµ
lim n n
1
by (C).
gM dµ −→ 0, the proof of this proposition is complete.
M
Theorem 5. Let (X, F, µ, T ) be an ergodic dynamical system and f ∈ L log L, f ≥
0. Then we have
∗
f (T k x) f (T n x) <∞
= sup n ·
(a)
k
n∗ 1
n
1,∞ 1
where
(b)
(c)
n∗
f (T x)
n∗
is for µ a.e. x a decreasing rearrangement of the sequence
f (T n x)
.
n
Z
Nn (f )(x)
lim
= f dµ, µ a.e.
n→∞
n
p 1/p Z
∞
X
f (T n x
lim (p − 1)
= f dµ, µ a.e.
p→1+
n
n=1
Proof. First we can make the following observations:
For all measurable sets A we have
1A (T k x) #{k :
= sup
k
t>0
1,∞
1A (T k x)
k
≥ 1/t}
t
= sup
#{k :
1A (T k x)
k
n
n
Nn (1A )(x)
n
n
= N ∗ (1A )(x).
(15)
= sup
Because of the maximal inequality for the ergodic averages we have
(16)
µ{x : N ∗ (1A )(x) > λ} ≤
1
· µ(A)
λ
for all λ > 0.
(Note that N ∗ (1A )(x) ≤ 1, hence for all p ≥ 1 we also have
µ{x : N ∗ (1A )(x) > λ} ≤
(17)
1
· µ(A)).
λp
For all positive real numbers y we have:
(18)
i
i
y i+1 = y i+1 ·
1
(i + 1)1/i+1
y(i + 1)1/i
i+
≤
(i + 1)
(i + 1)2
(i + 1)1/i+1
≥ 1/n}
26
I. Assani
(apply the inequality ab ≤
ap
p
+
bq
q ,
for a = y i/i+1 · (i + 1)1/i+1 , b =
p
p = i+1
i and q = p−1 = i + 1).
We proceed now with the proof of Theorem 5 (a).
We take f ∈ L log L and denote by Ai the set
1
,
(i+1)1/i+1
Ai = {2i ≤ f < 2i+1 }.
We have
N ∗ (f ) ≤ N ∗
X
∞
2i+1 1A )
≤
i=1
∞
X
N ∗ (2i+1 1A ))
i=1
∞
X
=2
2i · N ∗ (1A ).
i=1
By taking the integral with respect to the measure µ we get
kN ∗ (f )k1 ≤ 2
∞
X
2i kN ∗ (1Ai )k1
i=1
Using (17) we get
p
sup[t · µ{x : n∗ (1Ai )(x) > t}]
(p − 1) t>0
p
· (µ(Ai ))1/p
for all p, 1 ≤ p < ∞.
≤
(p − 1)
kN ∗ (1Ai )k1 ≤
p
((17) is combined with the inequality kgkL1 ≤ (p−1)
supt>0 [tµ{x : |g(x)| > t}1/p ].)
Going back to the evaluation of kN ∗ (f )k1 we get
kN ∗ (f )k1 ≤ 2
∞
X
i=1
=2
∞
X
2i
(i + 1/i)
(µ(Ai ))1/i+1
1/i
2i (i + 1)(µ(Ai ))i/i+1
i=1
=2
∞
X
((2i (i + 1))i+1/i µ(Ai ))i/i+1 .
i=1
Applying (18) to each term ((2i (i + 1))i+1/i µ(Ai ))i/i+1 we get
∗
kN (f )k1 ≤ 2
∞
X
[(2i (i + 1))i+1/i · (µ(Ai ))
i=1
≤4
∞
X
[2i iµ(Ai ) · (1 + i)2/i +
i=1
≤ 12 ·
∞
X
i=1
Z
12
≤
[
ln 2
[2i iµ(Ai ) +
1
]
(i + 1)2
1
]
(i + 1)2
f log f dµ + 1].
1
(i + 1)1/i
i+
]
i+1
(i + 1)2
Convergence of the p-Series for Stationary Sequences
27
Thus we have proved the following inequality
(19)
kN ∗ (f )k1 ≤
Z
12
[
ln 2
f log f dµ + 1]
for all f ≥ 0, f ∈ L log L.
This clearly ends the proof of Theorem 5 (a).
It remains to show (b). Our goal is to prove that for f ≥ 0, f ∈ L log L
lim ∗
N (f − f ∧ n) = 0 a.e.
n
(20)
Using (19) we have for all t > 0,
kN ∗ (t(f − f ∧ n))k1 ≤
Z
12
[ (t(f − f ∧ n)) log(t(f − f ∧ n))dµ + 1]
ln 2
for all f ≥ 0, f ∈ L log L.
This last inequality gives us
kN ∗ (f − f ∧ n)k1 ≤
Z
1
12
[ (f − f ∧ n) log(t(f − f ∧ n)]dµ + ].
ln 2
t
At the expense of taking a subsequence, we derive from it
lim kN ∗ (f − f ∧ nk )k1 ≤
k
12 1
· .
ln 2 t
∗
Then we easily get lim
n RN (f − f ∧ n) = 0 a.e. This proves (20).
As limn Nk (fk ∧n) = f ∧ ndµ, because f ∧ n is clearly bounded we have
Nk (f )
Nk (f ∧ n) Nk (f − f ∧ n)
Nk (f ∧ n)
≤
=
+
k
k
k
k
and after taking the limits we obtain
Z
f ∧ ndµ ≤
lim Nk (f )
Nk (f )
Nk (f ∧ n)
≤ lim
≤ lim
+ N ∗ [f − f ∧ n]
k
k
k
k
Z
= f ∧ ndµ + N ∗ [f − f ∧ n].
Finally, by taking the lim inf with respect to n we can conclude that
lim
k
Nk (f )
=
k
Z
f dµ a.e.
This proves Theorem 5 (b). Theorem 5 (c) now follows easily from Lemma 1. This
ends the proof of Theorem 5.
28
I. Assani
Corollary 6. Let (X, F, µ, T ) be an ergodic dynamical system and f ∈ L log L, f ≥
0. Then there exists an absolute constant C such that
p 1/p X
Z
∞ n
x)
f
(T
1/p
≤ C[ f log f dµ + 1]
sup (p − 1)
1<p<∞
n
n=1
1
Proof. By Theorem 5 we know that
n∗
sup n · f (T x) < ∞.
n
∗
n
1
As for µ a.e. x , for each p, we have
p 1/p X
1/p
∗
∞ ∞
X
f (T n x)
f (T n x)
p
≤ sup n ·
1/n
,
·
(p
−
1)
(p − 1)1/p
n
n∗
n
n=1
n=1
the corollary follows easily.
Remark.
1) One can see that the limit when p tends to ∞ of the p-Series is equals to
n
supn f (Tn x) . This is the reason why we only focus on the existence of the
limit when p tends to 1+.
2) We proved in [A2] that if N ∗ (f )(x) is a.e finite for all functions f ∈ L1+
then M2∗ (f )(x) is also a.e finite for all functions f ∈ L1 .
3) The results obtained in this note can be extended to increasing sequences of
integers (pn )n . The corresponding maximal function to consider is simply
f (T pn )(x)
k
k1,∞ .
n
To illustrate this we have the following Proposition.
Proposition 7. Let (X, F, µ, T ) be an ergodic dynamical system , p a fixed positive
real number 1 < p < ∞ and (pn )n an increasing sequence of positive integers.
Consider the following statements
f (T pn x) < ∞ a.e. for all f ∈ Lp+ .
(a)
n
1,∞
(b)
(c)
(d)
sup k
p−1
k
p
∞ X
f (T pn x)
n=k
sup
N
n
< ∞ a.e. for all f ∈ Lp+ .
N
1 X
f (T pn x) < ∞ a.e. for all f ∈ Lp+ .
N n=1
N
1 X
f (T pn x) < ∞ a.e. for all f ∈ L(p, 1).
sup
N N n=1
Then we have the following implications: (a) implies (d), (b) is equivalent to (c),
(b) implies (a) and (c) implies (d).
Convergence of the p-Series for Stationary Sequences
29
Proof. The implications (c) implies (b) and (b) implies (a) can be proved the same
way we did in [A1] for the usual Cesaro averages. (See the proof of Theorem 3 part
b) in [A1]). The implication (c) implies (d) is a direct consequence of the structure
of L(p, q) spaces as shown in [SW].
It remains to prove the implications (a) implies (d) and (b) implies (c). For (a)
implies (d), we can notice that (a) implies the existence of a finite constant Cp such
that for all f ∈ Lp+
Z
f (T pn x Cp
|f |p dµ
(21)
µ{x : >
λ}
≤
for all λ > 0.
n
λp
1,∞
In the particular case of f = 1A , (21) will give us the following
(22)
N
Cp
1 X
µ{x : sup
1A (T pn x) > λ} ≤ p µ(A)
N
λ
N
n=1
because
for all λ > 0
N
1A (T pn x) 1 X
=
sup
1A (T pn x).
n
N
N
1,∞
n=1
As this inequality is valid for all measurable sets A, we can conclude that the
maximal operator
N
1 X
∗
M (f )(x) = sup
f (T pn x)
N N n=1
is of restricted weak type (p,p) (see [SW]). In other words, the maximal operator
M ∗ maps the characteristic function of any measurable set A from L(p, 1) into
L(p, ∞). It is shown in [SW] that the nature of the maximal operator M ∗ and the
existence of an equivalent norm on L(p, ∞), making it a Banach space, M ∗ maps
continuously all functions f ∈ L(p, 1) into L(p, ∞). This means the existence of a
finite constant Cp such that for all f ∈ L(p, 1),
(23)
µ{x : sup
N
N
Cp
1 X
p
f (T pn x) > λ} ≤ p kf kp,1
N n=1
λ
for all λ > 0.
From this we can clearly derive (d).
The implication (b) implies (c) can be obtained by summation. For f ∈ Lp+ let
us denote by Cx the finite constant which dominates the sup on k. Then for each
k, we have
p
2k X
f (T pn x)
< Cx .
k p−1
2k
n=k
This implies the inequality
sup
k
2k
1X
p
f (T pn x) < 2p Cx .
k
n=k
From this we can derive by convexity the uniform boundedness of the averages
P2k
1
pn
x). This property allows us to obtain (c) without difficulty.
n=k f (T
k
Remark. It would be interesting to know if (a) implies (c) for any increasing
sequence pn and any p, 1 ≤ p < ∞ . For p = 1, pn = n we already mentioned that
(a) implies (c) (see [A2]).
30
I. Assani
References
[A1]
I. Assani, Strong laws for weighted sums of iid random variables, Duke Math. J. 88 (1997),
217–246.
[A2] I. Assani, Return times and Birkhoff theorems in L1 of product measures, Preprint.
[B]
D. Burkholder, Successive conditional expectations of an integrable function, AM St 33
(1962), 887–893.
[JOP] B. Jamison, S. Orey, and W. Pruitt, Convergence of weighted averages of independent
random variables, Z. Wahrheinlichkeitstheorie Verw. Gebiete. 4 (1965), 40–44.
[SW] E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton
University Press, Princeton, NJ, 1971.
Department of Mathematics, University of North Carolina at Chapel Hill, Chapel
Hill, NC 27599
assani@math.unc.edu
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